The Laws of Indices Explained
Tom Rocks Maths intern Laura Bradby explains where the laws of indices come from and how negative, zero and fractional powers work.
When we first learn about powers, we are usually told that only whole numbers can be used as an index, but this is an over-simplification. By deriving a series of rules or laws for indices, we can in fact calculate zero, negative and fractional powers.
The first law states that a^b x a^c = a^(b+c).
The second law states that (a^b)^c = a^(bc).
Using these laws, we can make a ‘sensible’ definition for zero powers, which must equal 1 for our laws to make sense. This then leads to a definition of negative powers as 1 divided by the positive power. Finally, we derive a rule for fractional powers where the denominator represents a root, and the numerator means to raise the root to that power.
Produced by TRM intern Laura Brady – a second year Mathematics student at the University of Oxford – with assistance from Dr Tom Crawford. Tom is an Early-Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-c...
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